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Chapter 8: Problem 37

Determine the surface area of the object described. Use 3.14 for \(\pi\) when necessary. A cube with each side \(4 \mathrm{cm}\) long

Short Answer

Expert verified
96 \text{cm}^2

Step by step solution

01

Identify the formula for the surface area of a cube

The surface area of a cube can be calculated using the formula: \[ \text{Surface Area} = 6a^2 \]where \( a \) is the length of one side of the cube.
02

Substitute the side length into the formula

Given that each side of the cube is 4 cm long, substitute \( a = 4 \text{cm} \) into the formula: \[ \text{Surface Area} = 6 \times (4\text{cm})^2 \]
03

Calculate the exponent first

Calculate \( 4^2 \): \[ 4^2 = 16 \]So, the equation now is: \[ \text{Surface Area} = 6 \times 16 \text{cm}^2 \]
04

Multiply to find the total surface area

Now multiply 6 by 16: \[ \text{Surface Area} = 96 \text{cm}^2 \]
05

Final surface area

The total surface area of the cube is: \[ 96 \text{cm}^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface Area Calculation
Calculating the surface area of a shape helps us understand how much space its exterior covers. For a cube, the surface area involves all six identical square faces.

The general formula for the surface area of a cube is given by: \[ \text{Surface Area} = 6a^2 \] Here, each side of the cube is denoted by \(a\). This formula multiplies the area of one square face by 6, since a cube has six faces.

When using the formula, start by squaring the side length, then multiply by 6.
  • Step 1: Identify the side length \(a\). For example, if \(a = 4 \text{cm}\).
  • Step 2: Compute \(a^2\), which is \(4^2 = 16 \text{cm}^2\).
  • Step 3: Multiply this result by 6 to get the total surface area: \(6 \times 16 \text{cm}^2 = 96 \text{cm}^2\).
Knowing how to calculate surface area is essential in geometry and real-world applications like wrapping gifts or painting a box.
Geometry Formulas
Geometry involves understanding and using different mathematical formulas to calculate dimensions like area, volume, and surface area.

Some key geometry formulas include:
  • Area of a square: \(A = a^2\)
  • Volume of a cube: \(V = a^3\)
  • Surface area of a sphere: \(4\pi r^2\)
  • Volume of a cylinder: \(\pi r^2h\)
  • Surface area of a cube: \(6a^2\)
Each formula is crucial because it helps solve different types of problems. For instance, while the surface area of a cube helps understand its exterior coverage, its volume formula \(a^3\) describes how much space the cube occupies internally.

Remembering and applying these basic formulas will give you a solid foundation in geometry, making it easier to tackle more complex problems!
Exponentiation
Exponentiation is a mathematical operation involving two numbers, the base and the exponent. It is written as \(a^b\), where \(a\) is the base, and \(b\) is the exponent.

Here's how it works: \[ a^b = a \times a \times \text{...} \times a \] (multiplying 'a' by itself 'b' times)

For example:
  • \(4^2\) means \(4 \times 4 = 16\)
  • \(3^3\) means \(3 \times 3 \times 3 = 27\)
  • \(2^4\) means \(2 \times 2 \times 2 \times 2 = 16\)
Exponentiation is fundamental in geometry formulas, like calculating the area of a square \(A = a^2\). In our cube surface area problem, finding \(4^2 = 16\) was an essential step before multiplying by 6 to determine the total surface area.

Mastering exponentiation can make complex calculations more manageable and is a critical skill in higher-level math and science disciplines.

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Most popular questions from this chapter

Convert the units of mass. $$0.003 \mathrm{kg}=_____ g$$

Cliff drives his children to their sports activities outside of school. When he drives his son to baseball practice, it is a \(6-\mathrm{km}\) round trip. When he drives his daughter to basketball practice, it is a \(1800-\mathrm{m}\) round trip. If basketball practice is 3 times a week and baseball practice is twice a week, how many kilometers does Cliff drive?

In the Expanding Your Skills of Section \(8.1,\) we converted U.S. Customary units of area. We use the same procedure to convert metric units of area. This procedure involves multiplying by two unit ratios of length. Example: Converting area Convert \(1000 \mathrm{mm}^{2}\) to square centimeters. $$\text { Solution: } \frac{1000 \mathrm{mm}^{2}}{1} \cdot \frac{1 \mathrm{cm}}{10 \mathrm{mm}} \cdot \frac{1 \mathrm{cm}}{10 \mathrm{mm}}=\frac{1000 \mathrm{mm}^{2}}{1} \cdot \frac{1 \mathrm{cm}^{2}}{100 \mathrm{mm}^{2}}=\frac{1000 \mathrm{cm}^{2}}{100}=10 \mathrm{cm}^{2}$$ convert the units of area, using two factors of the given unit ratio. $$65,000,000 \mathrm{m}^{2}=\quad \mathrm{km}^{2}$$ $$\left(\text { Use } \frac{1 \mathrm{km}}{1000 \mathrm{m}}\right)$$

In one day, Stacy gets \(600 \mathrm{mg}\) of calcium in her daily vitamin, \(500 \mathrm{mg}\) in her calcium supplement, and \(250 \mathrm{mg}\) in the dairy products she eats. How many grams of calcium will she get in one week?

Convert the units of mass. $$2011 \mathrm{g}= _____ kg$$
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